Subgroups — #79542

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#5127	C₂	1	2	yes	maximal, complemented
4	#5126	C₂	1	2	yes	maximal, complemented
5	#5012	C₂	1	1	yes	maximal, characteristic, complemented
6	#4835	C₂	1	1	yes	maximal, characteristic, complemented
7	#4834	C₂	1	1	yes	maximal, characteristic, complemented
8	#4728	C₂	1	2	yes	maximal, complemented
9	D₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#4685	C₂	1	1	yes	maximal, characteristic, complemented
12	#2974	—	3	1	no	maximal, Sylow
13	C₂×D₃×D₄	C₄	1	1	yes	characteristic
14	C₂²×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
16	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
17	D₄×Dic₃	C₄	1	1	yes	characteristic
18	C₄×#67	C₂²	1	2	yes	—
19	C₂×#208	C₂²	1	2	yes	complemented
20	C₂×D₃×C₈	C₂²	1	2	yes	complemented
21	C₂×D₃×C₈	C₂²	1	2	yes	complemented
22	#893	C₄	1	1	yes	characteristic
23	D₃×C₄⋊C₄	C₄	1	1	yes	characteristic
24	#67⋊C₄	C₄	1	2	yes	—
25	D₃×C₂²⋊C₄	C₄	1	2	yes	—
26	C₄×D₁₂	C₂²	1	1	yes	characteristic
27	C₄²×D₃	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
31	#830	C₂²	1	2	yes	complemented
32	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
37	#815	C₂²	1	2	yes	complemented
38	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
39	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₄×D₄	—	3	1	no	—
43	#436	—	3	1	no	—
44	#436	D₃	1	1	yes	characteristic, complemented
45	#435	—	3	2	no	—
46	#435	—	3	2	no	—
47	C₈×D₄	—	3	1	no	—
48	C₈×D₄	—	3	1	no	—
49	#432	—	3	1	no	—
50	#431	—	3	1	no	—
51	#407	—	3	2	no	—
52	C₂×C₂²⋊C₈	—	3	2	no	—
53	C₂³×D₃	C₂×C₄	1	2	yes	—
54	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂×#67	C₂×C₄	1	2	yes	—
57	C₂²×Dic₃	C₂×C₄	1	2	yes	—
58	D₃×D₄	—	2	4	no	—
59	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃	—	4	2	no	—
61	C₂×C₄×D₃	—	4	2	no	—
62	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
65	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
66	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₃	C₂³	1	2	yes	—
68	C₃×M₁₆	—	4	2	no	—
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
74	C₃×C₄²	C₂³	1	1	yes	characteristic
75	#217	C₂×C₄	1	2	yes	—
76	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
77	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
78	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
79	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
80	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₃	C₂³	1	1	yes	characteristic
83	#208	—	4	2	no	—
84	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂³	1	2	yes	—
87	D₃×C₈	—	4	2	no	—
88	D₃×C₈	—	4	2	no	—
89	C₂²×D₄	—	3	1	no	—
90	C₂³×C₄	—	3	2	no	—
91	C₂×M₁₆	—	3	2	no	—
92	C₂×M₁₆	C₂×D₃	1	2	yes	complemented
93	C₂²×C₈	—	3	2	no	—
94	C₂²×C₈	—	3	2	no	—
95	C₄×D₄	—	3	1	no	—
96	C₄×D₄	—	3	2	no	—
97	C₄×D₄	—	3	1	no	—
98	C₄×D₄	—	3	2	no	—
99	C₄×D₄	—	3	1	no	—
100	C₄×D₄	C₂×D₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	3	1	no	—
102	C₂×C₂²⋊C₄	—	3	2	no	—
103	C₂×C₄²	—	3	1	no	—
104	C₄⋊C₈	—	3	1	no	—
105	C₄⋊C₈	—	3	1	no	—
106	C₄⋊C₈	—	3	1	no	—
107	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
108	C₂²⋊C₈	—	3	2	no	—
109	C₂²⋊C₈	—	3	1	no	—
110	C₂²⋊C₈	—	3	2	no	—
111	C₂²⋊C₈	—	3	1	no	—
112	C₂²⋊C₈	—	3	2	no	—
113	C₂²⋊C₈	—	3	2	no	—
114	C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
115	#97	—	3	2	no	—
116	C₄×C₈	—	3	1	no	—
117	C₄×C₈	C₂×D₃	1	1	yes	characteristic, complemented
118	C₂³×C₃	C₂²×C₄	1	2	yes	—
119	C₂²×D₃	—	2	4	no	—
120	C₂²×D₃	—	2	4	no	—
121	C₂²×D₃	—	4	2	no	—
122	C₂²×D₃	C₂²×C₄	1	2	yes	—
123	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
124	C₃×D₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
129	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
131	#67	—	2	4	no	—
132	C₂×Dic₃	—	4	2	no	—
133	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×Dic₃	C₂²×C₄	1	2	yes	—
137	D₁₂	M₁₆	1	4	yes	complemented
138	C₄×D₃	—	2	2	no	—
139	C₄×D₃	—	2	2	no	—
140	C₄×D₃	—	2	2	no	—
141	C₄×D₃	—	2	2	no	—
142	C₄×D₃	—	2	1	no	—
143	C₄×D₃	—	2	1	no	—
144	C₄×D₃	—	4	2	no	—
145	C₄×D₃	—	4	2	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	2	1	no	—
149	C₃⋊C₈	—	2	1	no	—
150	C₃⋊C₈	—	2	2	no	—
151	C₃⋊C₈	—	2	1	no	—
152	C₂⁴	—	3	2	no	—
153	C₂×D₄	—	3	2	no	—
154	C₂×D₄	—	3	1	no	—
155	C₂×D₄	—	6	4	no	—
156	C₂×D₄	C₄×D₃	1	1	yes	characteristic
157	C₂²×C₄	—	3	2	no	—
158	C₂²×C₄	—	3	1	no	—
159	C₂²×C₄	—	3	1	no	—
160	C₂²×C₄	—	3	1	no	—
161	C₂²×C₄	—	3	2	no	—
162	C₂²×C₄	—	3	2	no	—
163	C₂²×C₄	—	3	2	no	—
164	C₂²×C₄	—	12	2	no	—
165	C₂²×C₄	—	12	2	no	—
166	C₂²×C₄	C₂²×D₃	1	2	yes	—
167	M₁₆	—	4	2	no	—
168	M₁₆	—	12	2	no	—
169	C₂×C₈	—	3	1	no	—
170	C₂×C₈	—	3	1	no	—
171	C₂×C₈	—	3	2	no	—
172	C₂×C₈	—	12	2	no	—
173	C₂×C₈	—	12	2	no	—
174	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₃	1	2	yes	—
177	C₄⋊C₄	—	3	2	no	—
178	C₄⋊C₄	—	3	1	no	—
179	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	3	2	no	—
181	C₂²⋊C₄	—	3	2	no	—
182	C₂²⋊C₄	—	3	2	no	—
183	C₂²⋊C₄	C₄×D₃	1	2	yes	—
184	C₄²	—	3	1	no	—
185	C₄²	—	3	1	no	—
186	C₄²	—	3	1	no	—
187	C₄²	C₂²×D₃	1	1	yes	characteristic
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
192	C₂×D₃	—	2	1	no	—
193	C₂×D₃	—	2	2	no	—
194	C₂×D₃	—	2	1	no	—
195	C₂×D₃	—	2	2	no	—
196	C₂×D₃	—	4	8	no	—
197	C₂×D₃	C₂×M₁₆	1	4	yes	—
198	C₂×D₃	#131	1	2	yes	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	C₂×M₁₆	1	2	yes	—
202	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
203	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
204	Dic₃	—	2	1	no	—
205	Dic₃	—	2	2	no	—
206	Dic₃	—	2	1	no	—
207	Dic₃	—	2	1	no	—
208	C₂³	—	3	2	no	—
209	C₂³	—	3	1	no	—
210	C₂³	—	6	4	no	—
211	C₂³	—	6	4	no	—
212	C₂³	—	12	2	no	—
213	C₂³	C₂×C₄×D₃	1	2	yes	—
214	D₄	—	2	2	no	—
215	D₄	—	3	4	no	—
216	D₄	—	6	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	3	1	no	—
220	C₂×C₄	—	3	1	no	—
221	C₂×C₄	—	3	1	no	—
222	C₂×C₄	—	3	2	no	—
223	C₂×C₄	—	6	2	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	1	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	—	6	1	no	—
229	C₂×C₄	—	12	2	no	—
230	C₂×C₄	—	12	2	no	—
231	C₂×C₄	—	12	2	no	—
232	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	6	1	no	—
240	C₈	—	6	2	no	—
241	C₈	—	6	1	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	#518	1	1	yes	characteristic
244	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
245	C₂×C₃	C₂×#131	1	1	yes	characteristic
246	D₃	—	2	2	no	—
247	D₃	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	3	4	no	—
252	C₂²	—	3	2	no	—
253	C₂²	—	6	1	no	—
254	C₂²	—	6	2	no	—
255	C₂²	—	6	1	no	—
256	C₂²	—	6	2	no	—
257	C₂²	—	12	8	no	—
258	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	6	1	no	—
262	C₄	—	6	1	no	—
263	C₄	—	6	1	no	—
264	C₄	—	6	2	no	—
265	C₄	D₃×M₁₆	1	2	yes	—
266	C₄	#1004	1	1	yes	characteristic
267	C₄	#1009	1	1	yes	characteristic
268	C₃	#2974	1	1	yes	characteristic, complemented, Sylow
269	C₂	—	4	2	no	—
270	C₂	—	6	2	no	—
271	C₂	—	6	4	no	—
272	C₂	#5545	1	1	yes	characteristic
273	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
274	C₂	D₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#436	C₂
prime radical	C₂×D₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#436	D₃
hypocenter	C₃	#2974
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#436	C₂	semidirect
3. #5127	C₂	semidirect
4. #5126	C₂	semidirect
5. #5012	C₂	semidirect
6. #4835	C₂	semidirect
7. #4834	C₂	semidirect
8. #4728	C₂	semidirect
9. D₃×C₂²⋊C₈	C₂	semidirect
10. C₈×D₁₂	C₂	semidirect
11. #4685	C₂	semidirect
15. C₂×C₃×M₁₆	C₂²	semidirect
19. C₂×#208	C₂²	semidirect
20. C₂×D₃×C₈	C₂²	semidirect
21. C₂×D₃×C₈	C₂²	semidirect
28. C₃×C₄⋊C₈	C₂²	semidirect
29. C₃×C₂²⋊C₈	C₂²	semidirect
30. C₃×C₄×C₈	C₂²	semidirect
31. #830	C₂²	semidirect
35. (C₂×D₃)⋊C₈	C₂²	semidirect
36. (C₂×D₃)⋊C₈	C₂²	semidirect
37. #815	C₂²	semidirect
38. Dic₃⋊C₈	C₂²	semidirect
39. Dic₃⋊C₈	C₂²	semidirect
40. C₄⋊(C₃⋊C₈)	C₂²	semidirect
41. C₄×C₃⋊C₈	C₂²	semidirect
44. #436	D₃	semidirect
92. C₂×M₁₆	C₂×D₃	semidirect
107. C₄⋊C₈	C₂×D₃	semidirect
114. C₂²⋊C₈	C₂×D₃	semidirect
117. C₄×C₈	C₂×D₃	semidirect
137. D₁₂	M₁₆	semidirect
268. C₃	#2974	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×D₃×D₄	2
C₃×#436	2
#5127	2
#5126	2
#5012	2
#4835	2
#4834	2
#4728	2
D₃×C₂²⋊C₈	2
C₈×D₁₂	2
#4685	2
#2974	3